Unicidad de soluciones largas

UNIVERSIDAD COMPLUTENSE DE MADRID

FACULTAD DE CIENCIAS MATEMÁTICAS

Departamento de Matemática Aplicada

TESIS DOCTORAL

Unicidad de soluciones largas

(Uniqueness of large solutions)

MEMORIA PARA OPTAR AL GRADO DE DOCTOR PRESENTADA POR

Luis Maire Martín

Director

Julián López-Gómez

Madrid, 2018

UNIVERSIDAD COMPLUTENSE DE MADRID

Doctorado en Investigaci´on Matem´atica Facultad de Ciencias Matem´aticas Departamento de Matem´atica Aplicada

TESIS DOCTORAL

Unicidad de soluciones largas

(Uniqueness of large solutions)

Memoria para optar al t´ıtulo de Doctor Presentada por

Luis Maire Mart´ın

Director

Juli´an L´opez-G´omez

Acknowledgments

To begin this Thesis, the fairest thing I can do is to express my most sincere gratitude to everyone who has helped me during this process. Sometimes, one believes that has reached his aims exclusively due to his self skills, but this is a nonsense that leads us to forget the others. Thus, I would like to mention all members of the team behind me, to whom I owe this work.

First, I am fully grateful to Juli´an L´opez-G´omez, for all the effort, dedication and trust he has placed in me. The greatest virtue of a professor is to generate interest amongst his students, and this is exactly what I experienced when I attended his lectures during my Degree on Mathematics. I became his pupil after this phase, starting off a wonderful and fruitful period to me, in which I have learned much more than I could have ever imagine. Juli´an, for driving me to the highest mathematical level, thanks!

Thanks very much to Pierpaolo Omari, who has helped me a lot when I was staying at Trieste. It has been a fabulous period that I will never forget.

To my father, Pepe Maire, and my mother, Adita Mart´ın. Every day I realize how lucky I am for having such an incredible parents. Thank you very much for all, I love you!

Actually, my fortune is double, since I have another two parents as wonderful as these. I warmly thank to Maika Arag´on and Ramon Irigoyen their infinite generosity and kindness for opening their house to me. This thesis would have been very difficult without your help. To my lovely sister, Susana Maire, who is always up for anything. I would like also thank to Mario L´opez. I hope that Valle brings to both of you all the happiness of the world during the rest of your lives. I am sure that you will become exceptional parents! My best wishes, from the deepest of my heart, will be always with you!

It could not miss a very special mention to my grandmothers, Juana Cabezas and Ma Cruz de los Pinos. Unfortunately, Ma Cruz left us before I could finish this Thesis. I also thank to all my uncles and cousins, specially to Bel´en Cano and Jos´e MaMart´ın.

Talking about my family, it is inevitable remember my village, Villarejo del Valle, where one founds very nice and funny people, as Luis and Ovidio Bermejo, who have given me tons of lifts to Madrid. For the same reason, I thank to Jos´e Ma de los Pinos, Elena de la Fuente, Miguel A. Rey, Matilde P´erez and Jes´us Bermejo.

Going back to Madrid, I thank to Jorge Gonz´alez and Roberto Tom´e, who have helped me a lot throughout the doctorate, and to Eduardo Fern´andez, Luis Izquierdo and Javier Pecharrom´an.

4

I also send a very warm acknowledgment to my closest friends, for all great times of happiness and enjoyment: Jes´us Iglesias, Marcos Jim´enez, Juanma Montoya, Victor Rodr´ıguez, Cr´ıstofer Hern´andez and Kike Lokomotoro. And to Paquita Doblas, Mariano Montoya, Soledad Gonz´alez, Santos Jim´enez, Jacinto Rodr´ıguez and Julia Fern´andez, for their great hospitality.

To conclude these acknowledgements, I dedicate the most special of all to Ana Irigoyen, my love, who has tremendously supported me during all these years. Ana, thanks a lot for making me the happiest man of the world!

Trieste, 28 April 2017. Luis Maire

Agradecimientos

Para comenzar esta tesis, lo m´as justo y sensato es expresar mi gratitud a todos los que, de un modo u otro, me han ayudado a realizarla. A veces, es f´acil dejarse llevar por la idea absurda de que nuestros logros se deben en exclusivo a las propias facultades y, en consecuencia, olvidamos a los dem´as. Por tanto, quisiera nombrar a todos los miembros de ese gran equipo que se encuentra a mis espaldas, a quienes debo enteramente este trabajo.

En primer lugar, quiero agradecer a Juli´an L´opez-G´omez todo el esfuerzo, dedicaci´on y confianza que ha depositado en m´ı. La mayor virtud de un profesor es saber despertar el inter´es de sus alumnos hacia lo que les ense˜na. Yo tuve la suerte de experimentarlo cuando fui alumno de Juli´an durante el grado y el m´aster. Tras este periodo, comenzamos a trabajar como alumno y director de tesis, inici´andose para m´ı una etapa tremendamente fruct´ıfera, en la que he aprendido mucho m´as de lo que pod´ıa haber imaginado. Juli´an, por tu entrega incondicional a este proyecto, ¡gracias!

Muchas gracias a Pierpaolo Omari por haberme dedicado su tiempo y hacer que mi estancia en Trieste haya sido tan agradable. Ha sido un periodo maravilloso que jam´as olvidar´e.

A mi padre, Pepe Maire, y mi madre, Adita Mart´ın. Cada d´ıa soy m´as consciente de lo afortunado que soy por tener unos padres tan incre´ıbles. Gracias por todo el amor que me hab´eis dado. ¡Os quiero!

La fortuna de tener unos padres as´ı es incalculable, pero en mi caso es todav´ıa mayor al poder sumar otros dos padres igual de maravillosos. Quiero agradecer de todo coraz´on a Maika Arag´on y Ram´on Irigoyen su infinita generosidad al abrirme las puertas de su casa. Esta tesis habr´ıa sido muy dif´ıcil sin vuestra ayuda.

A mi querid´ısima hermana, Susana Maire, que siempre est´a dispuesta a todo por m´ı. Tamb´ıen quiero dar las gracias a Mar´ıo L´opez. Espero que Valle os traiga toda la felicidad del mundo durante el resto de vuestras vidas. ¡Vais a ser unos padres geniales! Os deseo lo mejor.

No pod´ıa faltar una menci´on muy especial a mis abuelas, Juana Cabezas y Ma Cruz de los Pinos, quien, desgraciadamente, nos dej´o antes de que pudiese acabar esta tesis. Tambi´en quiero dar gracias a todos mis t´ıos y primos, especialmente a Bel´en Cano y Jos´e Ma Mart´ın. Y como mi familia es muy grande, tanto en tama˜no como en coraz´on, seguro que podr´an perdonarme aqu´ellos que no han sido nombrados.

Recordando a mis familiares, es inevitable pensar en mi pueblo, Villarejo del Valle,

6

donde adem´as de paisajes de ensue˜no, se pueden encontrar personajes tan desternillantes como generosos. Hablo de Luis y Ovidio Bermejo, que tan amablemente me han llevado a Madrid cientos de veces. Por lo mismo, agradezco a Jos´e Ma de los Pinos, Elena de la Fuente, Miguel A. Rey, Matilde P´erez y Jes´us Bermejo.

Volviendo a Madrid, quiero dar las gracias Jorge Gonz´alez y Roberto Tom´e, que tanto me han ayudado durante el doctorado, y a Eduardo Fern´andez, Luis Izquierdo y Javier Pecharrom´an.

A mis amigos, Jes´us Iglesias, Marcos Jim´enez, Juanma Montoya, Victor Rodr´ıguez, Cr´ıstofer Hern´andez y Kike Lokomotoro, con quienes comparto siempre los mejores mo-mentos de alegr´ıa y goce. Agradezco tambi´en a Paquita Doblas, Mariano Montoya, Soledad Gonz´alez, Santos Jim´enez, Jacinto Rodr´ıguez y Julia Fern´andez la calidez con la que siem-pre me reciben en su casa.

Para finalizar estos agradecimientos, dedico el m´as especial de todos a Ana Irigoyen, quien m´as me ha apoyado durante todos estos a˜nos. Mi vida, gracias por lo inmensamente feliz que me haces. Te quiero infinito.

En Trieste, a 28 de abril de 2017. Luis Maire

Contents

Summary 9

0.1 Introduction, motivation and methodology . . . 9

0.2 Content . . . 11

0.3 Conclusions . . . 17

Resumen 19 0.4 Introducci´on, motivaci´on y metodolog´ıa . . . 19

0.5 Contenido . . . 22

0.6 Conclusiones . . . 24

27 I Largesolutionsfortheequation 1 Multiplicityoflargesolutionsinonespatialdimension 29 1.1 The associated Cauchy problem . . . 31

1.2 Existence, uniqueness and multiplicity . . . 36

1.3 Counterexamples . . . 42

2 Uniquenessoflargepositivesolutions 45 2.1 The star-shaped case . . . 47

2.1.1 Proof of Theorem 2.2 . . . 49

2.2 The generalized star-shaped case . . . 52

2.2.1 Proof of Theorem 2.4 . . . 53

2.3 Uniqueness in smooth domains . . . 55

2.3.1 Proof of Theorem 2.5 . . . 58

61 II Largesolutionsofcooperativesystems 3 Introductiontocooperative-logisticsystems 63 3.1 Existence of large solutions . . . 64

8 CONTENTS

4 Theradiallysymmetriccase 71

4.1 Proof of Theorem 4.1 in caseΩ =BR(x0). . . 72

4.2 Proof of Theorem 4.1 in caseΩ =AR1,R2(x0). . . 74

5 Boundaryblow-upratesofthelargesolution 81 5.1 A natural way of finding out the blow-up rates . . . 86

5.2 Two pivotal technical results under radial symmetry . . . 89

5.3 Proofs of the main results . . . 97

5.3.1 Proof of Theorem 5.1 . . . 97

5.3.2 Proof of Theorem 5.4 . . . 102

Summary

0.1

Introduction, motivation and methodology

The principal aim of this doctoral thesis is to establish uniqueness results, as much general as possible, to the following diffusive logistic elliptic system,

       −∆ui=λiui+ n X j=1,j6=i aijuj−ai(x)fi(ui)ui in Ω, ui = +∞ on ∂Ω, 1≤i≤n, (0.1)

wheren ∈_N,Ωis a bounded subdomain ofRN,N ∈N, with regular boundary,λi ∈ R, aij >0are the coupling parameters andai∈ Cν( ¯Ω)for someν ∈(0,1)satisfiesai(x)>0, for allx∈Ωand1≤i, j ≤n. The singular boundary conditions should be understood in the sense that

lim

dist(x,∂Ω)↓0ui(x) = +

∞, 1≤i≤n.

Thus, this Thesis deals withlarge, or explosive, solutions. The model (0.1) is a general-ization of the diffusive logistic equation studied in Chapters 6, 7 and 8 of [48], to a system with linear cooperative coupling, becauseaij > 0for all1 ≤ i, j ≤ n, i 6= j. As far as concerns the single generalized logistic equation,

−∆u=λu−a(x)f(u)u,

there is a huge amount of literature. The most pioneering results go back to L. Bieber-bach [9] and H. Rademacher [69], who considered the equation∆u=euin two and three dimensions, respectively, and J. B. Keller [35] and R. Osserman [66], who dealt with the equation∆u =f(u)for some class of monotonef’s. Since then, studies on solutions of single equations with boundary blow-up have followed in many different ways. For exam-ple, establishing existence and uniqueness results for more general kinds of nonlinearities, as in [8, 19, 28, 14], for models with spatial heterogeneities, [73, 7, 18, 12, 13, 41, 45], for more general elliptic operators, [73, 16], or even considering domains with irregular boundary, [58, 59]. Another usual topic in the framework of large solutions is to establish the asymptotic boundary expansion of the solution, as in [2, 14, 3]. There are also some astonishing multiplicity results in the context of superlinear indefinite problems, as the ones

10 SUMMARY

of [42, 60, 55, 56]. Nevertheless, the literature on systems is substantially more reduced. A pioneering work dealing with (0.1) under Dirichlet boundary conditions in casen = 2 is [61, 62, 63], where it is studied the existence and the uniqueness of positive solutions. One of the first papers on large solutions for systems is [32], where the authors characterized the existence of large solutions to the classical diffusive symbiotic model of Lotka-Volterra, as well as the blow-up rates of each of the components of these singular solutions. More recently, [31] showed the existence and uniqueness of large solutions for a class of

au-tonomousreaction diffusion systems of cooperative type. As far as we know, the first paper

that shows the existence of a solution for (0.1), in casen= 2, is [4]. Therefore, the problem of the uniqueness for (0.1) remained utterly open. Actually, this problem is still open, even for the single equation.

Although the uniqueness has a great interest from a mathematical point of view, one can provide an important motivation that arises in the context of Population Dynamics. Let

D ⊂_RN_{,N ≤3, the inhabiting area of a species and denote byu=u(t, x)the population} density of the species, which varies with space,x ∈ D, and timet≥0. Let us asume the next hypotheses:

• The species is divided into ngroups, u1, . . . , un, such that each group cooperates with all others, in the sense that ifuigrows thenujalso grows, for every1≤i, j≤n,

j 6=i.

• Each ui spans randomly in the inhabiting areaD, with diffusion rate measured by

di >0,1≤i≤n.

• There exist some places in D with unlimited natural resources, while others have limited natural resources. This entails that the species will grow according to the Malthus law where the resources are unlimited, while it has a logistic growth in the complement. By the sake of simplicity, we suppose thatΩ⊂D¯is the zone where the natural resources are limited.

Keeping in mind the last assumptions, a possible mathematical model for the evolution of this species might given by

             ∂ui ∂t −di∆ui=λiui+ n X j=1,j6=i aijuj−bi(x)fi(ui)ui x∈ D, t >0 ui= 0 on ∂D, ui(0) =u0,i >0, (0.2)

where1≤i≤nandbi ∈ Cν( ¯D)satisfies

   bi(x)>0 if x∈Ω, b_i(x) = 0 if x6= Ω, 1≤i≤n.

0.2. CONTENT 11

Thus, we have combined in D both laws of population growth mentioned above. The Dirichlet homogeneous boundary conditions in (0.2) are imposed just by simplicity. They entail that every member reaching the boundary, dies. Of course, a very interesting problem would be to consider another boundary conditions.

What is the asymptotic behavior of the unique global solution of Problem (0.2)? The answer to this question is closely related to the problem of the uniqueness of (0.1), in the following way. Under some assumptions on the range of the parametersλi,1≤i ≤n, if we denote byLmin_andLmax_{theminimaland themaximalsolutions of (0.1), we have that}

Lmin≤lim inf

t→+∞u(t)≤lim sup_t→+∞ u(t)≤L

max_{, x∈Ω, (0.3)}

whileu(t) ↑ +∞ inD \Ω. Therefore, metasolutions do govern the dynamics of (0.2), within some open regions of the parametersλi. Grosso modo, a metasolution is a steady state of (0.2) equaling infinity somewhere. The reader is sent to [33, 4], or [48, Chapter 5], for a precise definition of metasolution. Summarizing, the uniqueness of the solution of (0.1) entailsLmin = Lmaxin (0.3), which characterizes the asymptotic behavior of the solutions of (0.2).

The most important technique used throughout this Thesis is the following comparison principle, derived from the theorem of characterization of the maximum principle of [54]. Letw1, w2 ∈

C2+ν_(∂Ω)n

such thatw1,i < w2,ifor every1 ≤ i≤ n. Then, the unique positive solution of        −∆ui =λiui+ n X j=1,j6=i aijuj −ai(x)fi(ui)ui in Ω, ui =wk,i>0 on ∂Ω, 1≤i≤n, (0.4)

throughout denoted by θ[Ω,wk], k = 1,2, satisfies θ[Ω,w1] < θ[Ω,w2]. Actually, the same

principle holds ifw2 = (+∞, . . . ,+∞)on∂Ω. In order to show this, it suffices applying

the previous result to

Ωε :={x∈Ω : dist(x, ∂Ω)> ε}

for sufficiently smallε >0and then lettingε↓0. Thanks to this comparison principle, we have achieved many of the theorems of this thesis. In particular, the results of Chapters 2, 3 and 4 are deduced by applying it in a number of rather sophisticated ways.

Besides the previous comparison principle, in Chapter 5 we have used one of the most usual uniqueness techniques: Establishing explicit formulas for theboundary blow-up rates

of the solutions of (0.1). Finally, we have adapted to our present context the localization method introduced in [41] for the single equation.

0.2

Content

The results established in this Thesis have been obtained by the author together with his superadvisor during 2015, 2016 and the first three months of 2017. Among them, those

12 SUMMARY

of [49, 50, 51] have already been published, while [52, 53] are in process of publication. We have ordered them not in chronological order, but according to the number of equations taken into consideration. The main contributions provided in each chapter of this Thesis are the following:

(1) The first chapter contains the results of [53], where we introduced the large solutions through the simplest model possible,

 



u00=f(u) t∈(−T, T),

u(±T) = +∞. (0.5)

The novelty of our analysis is that we do not impose any restriction on the sign off

andf0, as it is done in all previous studies, wheref ≥0in[0,∞), or for sufficiently largeu. This difference may change substantially the dynamics of (0.5). Indeed, the function T(x) := √1 2 Z +∞ x du q Ru x f , x:=u(0),

provides us with the maximal existence time of any solution ofu00 =f(u), when it blows up. The conditionf(u)≥0for largeuimplies

lim inf

x→+∞T(x) = 0,

which does not necessary happen without sign restrictions onf, as the counterexam-ples of the last section of this chapter show. We also establish a rather astonishing multiplicity result of large solutions from any given increasing positive functionf(u) that satisfies the Keller–Osserman condition, destroying the monotonicity off(u)on a compact set with arbitrarily small measure.

(2) Chapter 2 contains the results of [52] and it consists of three uniqueness theorems for the singular boundary value problem

 



−∆u=λu−a(x)f(u) in Ω,

u= +∞ on ∂Ω, (0.6)

where λ∈ _R,f ∈ C1_{[0,+∞),f(0) = 0,f}0 _{≥ 0 anda(x) > 0 for everyx ∈ Ω.}

As it is usual, to get existence of solutions to (0.6), one should assume thatf satisfies the following adaptation of the classical condition of Keller [35] and Osserman [66]:

(KO) For everyα >0there existsu∗=u∗(α)>0such that

I(u) := Z +∞ 1 dθ q Rθ 1(α f(ut) u −t)dt <+∞ for all u > u∗,

0.2. CONTENT 13

The first result of this chapter provides with some sufficient conditions for the unique-ness in star-shaped domains:

Theorem 0.1 SupposeΩis star-shaped,λ≥0andf satisfies the next property:

(C) There existsp >1such that

f(tu)≥tpf(u) for all t >1 and u >0. (0.7)

Moreover, assume that there existsη >0such that, for everyz∈∂Ω,

az+ x0−z |x0−z|t ≤az+ x0−z |x0−z|s if 0< t < s < η. (0.8)

Then,(0.6)has a unique positive solution.

The second theorem of Chapter 2 is an adaptation of Theorem 0.1 to cover the class of domains which can be represented as an star-shaped domain with m-star-shaped holes and it can be stated as follows:

Theorem 0.2 Suppose λ = 0in (0.6) andf satisfies (C). Moreover, assume that

there are an integerm ≥ 1andm+ 1star-shaped domains, Ωi, 0 ≤i ≤ m, with

∂ΩiLipschitz continuous, such that

¯ Ωi ⊂Ω0, Ω¯i∩Ω¯j =∅, 1≤i, j≤m, i6=j, and Ω = Ω0\ Ω¯1∪ · · · ∪Ω¯m . (0.9)

For every0 ≤i≤m, let denote byxithe (center) point with respect to whichΩi is

star-shaped. Finally, suppose that there existsη >0such that, for every1≤i≤m,

z0 ∈∂Ω0andzi ∈∂Ωi, a z0+ _|x_x0₀−₋z_z0_0|t ≤a z0+_|x_x0₀−₋z_z0_|s azi+ _|z_xi_i−₋x_zi_i|t ≤azi+ _|z_xi_i−₋x_zi_i|s    if 0< t < s < η, 1≤i≤m. (0.10)

Then,(0.6)has a unique positive solution.

As far as concernsΩ, any annular region satisfies the requirements of Theorem 0.2, as well as any ball,Ω0, perforated by finitely many closed disjoint balls,Ωi,1≤i≤m. The last theorem of Chapter 2 studies the uniqueness in smooth domains:

Theorem 0.3 LetΩ∈ C1 _{such that the uniform interior sphere property is satisfied} on ∂Ωandλ < σ[−∆,Ω]. Assume that, for everyz ∈ ∂Ω, there existsδ > 0such that|x−z|< δ, withx∈Ω, implies

14 SUMMARY

where nz stands for the outward unit normal vector to∂Ωatz. Moreover, suppose

thatf issuper-additivewith constantC ≥0, i.e., there existsC≥0such that

f(a+b)≥f(a) +f(b)−C for all a, b≥0. (0.12)

Then,under(KO), the singular boundary value problem(0.6)possesses a unique

pos-itive solution.

Although at first glance (0.11) might look a little bit strange, it just means thata(x)is non-increasing asxapproximates∂Ωalong parallel rays to the line passing through

zandz+nz. Naturally, it holds if eithera(x)is constant in a neighborhood of∂Ω, or ifa∈ C1_{( ¯Ω)and}

∂a ∂nz

(z)<0 for all z∈∂Ω.

The superadditivity property (0.12) goes back to Theorem 0.3 of [58], where Marcus and V´eron obtain uniqueness of large solutions of

−∆u+f(u) = 0 (0.13)

for domains whose boundary is locally the graph of a continuous function. This is an extremely weaker hypothesis on the regularity of∂Ωthan ours, although their proof needs f to be convex. Moreover, there is no linear term in (0.13), and no spatial heterogeneities can be incorporated to the model without some additional further work.

Theorem 0.3 is a new finding even in the autonomous case:

Corollary 0.4 SupposeΩ∈ C1_{satisfies the uniform interior sphere property on∂Ω} andλ < σ[−∆,Ω]. Assume that(0.12)and(KO)hold. Then, ifa(x) = 1for every

x∈Ω, the singular boundary value problem(0.6)has a unique positive solution.

(3) In the third chapter we introduce the system (0.1) and derive the existence of a mini-mal and a maximini-mal solution of (0.1), adapting the corresponding existence theorems of [48] and [4]. Precisely, we establish the comparison principle for cooperative sys-tems explained in the previous section, getting as a consequence that the mapping

m7→θ_{[Ω, ~m]}

is increasing, wherem~ := (m, . . . , m). Thus, the point-wise limit

θ_[Ω,∞](x) :=_mlim_→+∞θ[Ω, ~m](x), x∈Ω, (0.14)

is well defined, though it might be infinite somewhere in Ω. For this reason, it is natural to assume some Keller-Osserman condition for the system, as, for example, the existence of an increasingF ∈ C1_{[0,+∞)withF(0) = 0andf}

0.2. CONTENT 15

such thatF satisfies Condition (KO). Under the latest assumption, the limit given by (0.14) is finite inΩand provides us with the minimal solution of (0.1). Moreover, the maximal solution of (0.1) is given by

Lmax= lim

δ↓0θ[Ωδ,∞],

whereΩδ :={x∈Ω : dist (x, ∂Ω)> δ}.

(4) Chapter 4 discusses the main result of [49], where a radially symmetric counterpart of (0.1) is studied. Its main result stands as follows.

Theorem 0.5 Suppose thatΩis a ball or an annulus in(0.1),λi ≥0, and

a_i(x) =a_i(dist(x, ∂Ω)), 1≤i≤n,

are positive nonincreasing functions. Suppose the functiong(u) :=f(u)u satisfies

Condition (C)defined in(0.7). Then, Problem(0.1)has a unique positive solution.

Moreover, it is radially symmetric.

The proof of this theorem is based on a rather sophisticated use of the maximum prin-ciple for weakly coupled cooperative elliptic systems, without invoking to the bound-ary blow-up rates of the large solutions. This result is the first available uniqueness theorem in the literature forn-species cooperative systems.

(5) Lastly, in Chapter 5 we study the blow-up rates of the solutions of (0.1). The results summarized here were first established in [50] for the casen= 2, and later in [51] in the general case. The main result of this chapter provides us, for eachz∈∂Ω, with

αi(z), Ai(z)>0such that lim x→z x∈Ω ui(x) dist(x, ∂Ω)−αi(z) =Ai(z), 1≤i≤n, (0.15)

for any solution of (0.1), u = (u1, . . . , un), under some special assumptions on the terms fi andai. Concretely, we ascertain the boundary blow-up rates in case

fi(u) = upi−1 for some pi > 1, assuming that ai(x) behaves like a power near

∂Ω,not necessary with fixed rate, in the sense that there existbi, γi ∈ C(∂Ω), with

b_i(z)>0for allz∈∂Ωandγi≥0on∂Ω, such that

lim x→z x∈Ω,z∈∂Ω

a_i(x)

b_i(z)[dist(x, ∂Ω)]γi(z) = 1, 1≤i≤n. (0.16)

By [41], it is well known that, setting

µi(z) :=

γi(z) + 2

pi−1

16 SUMMARY

for every z ∈ ∂Ω, these µi(z)’s provide us with the blow-up rates on ∂Ω of the positive solutions of the uncoupled singular problems

   −∆ui=λi(x)ui−ai(x)upii inΩ, ui= +∞ on∂Ω, 1≤i≤n.

Letz∈∂Ωand suppose thenequations of (0.1) have been re-ordered so that 0< µn(z)≤µn−1(z)≤ · · · ≤µ1(z). (0.18)

Then, we have the next result, which is completely new ever in the very special case whenn= 2.

Theorem 0.6 Letz∈∂Ωsuch that(0.18)is satisfied and consider the next partition of the subscripts set

I+:={i∈ {1, . . . , n}: µi(z) + 2−µ1(z)>0}, I0 :={i∈ {1, . . . , n}: µi(z) + 2−µ1(z) = 0}, I−:={i∈ {1, . . . , n}: µi(z) + 2−µ1(z)<0}. Letk∈ {1, . . . , n}be such that

IM :={i∈ {1, . . . , n}: µi(z) =µ1(z)}={1, . . . , k}. Then, any positive solution of (5.1),u= (u1, . . . , un), satisfies(0.15)with

αi(z) :=      µi(z) if i∈I+∪I0, µ1(z) +γi(z) pi if i∈I−, and Ai(z) :=                                µi(z)(µi(z) + 1) b_i(z) 1 pi−1 if i∈I+,   1 bi(z) k X j=1 aij µ1(z)(µ1(z) + 1) bj(z) 1 pj−1   1 pi if i∈I−, A0,i if i∈I0,

whereA0,istands for the unique positive solution of the equation

bi(z)xpi−µi(z)(µi(z) + 1)x= k X j=1 aij µ1(z)(µ1(z) + 1) b_j(z) 1 pj−1 .

0.3. CONCLUSIONS 17

The most astonishing fact is that if the blow-up rates of the uncoupled system are sufficiently close between them, then the corresponding blow-up rates of (0.1) equal the uncoupled ones, without taking into account the size of the coupling coefficients. On the other hand, the blow-up rates of (0.1) can differ from the uncoupled blow-up rates when some of the uncoupled blow-up rate is far from the others. As our model introduces spatial heterogeneities, the uncoupled blow-up rates may vary throughout

∂Ω, by (0.16). Hence, the blow-up rates of (0.1) may behave like the uncoupled

ones on some places of ∂Ω, while may be affected by the coupled coefficients on

other locations. This surprising behavior was documented by the first time in [50]. No previous result on blow-up rates for n-species cooperative systems is available before [51].

0.3

Conclusions

Although problems with singular boundary conditions have an enormous difficulty, by the huge number of technicalities involved in their mathematical treatment, we have succeeded in getting a number of uniqueness theorems based on the comparison theorem. Hence, imposingaij > 0for all1 ≤ i ≤ nis imperative to carry out our analysis. Ac-tually, if some coupling coefficient, aij, becomes negative, then the maximum principle for cooperative systems [54] fails, and therefore, this comparison technique also fails. It would be pretty interesting to consider this kind of situations in the future. A good way to approach this problem would be through thequasi-cooperativecase, i.e. whenn = 2and

a12a21>0.

As a byproduct of our investigations, we are providing with some refinements and ex-tensions of the usual uniqueness techniques as well as some new ideas. Theorem 0.5 is an adaptation of a result of [46] to cover the class of cooperative systems with radial symmetry. This adaptation is certainly not trivial in the annular case, where an auxiliary construction is required. Theorems 0.1 and 0.2 are also inspired by the main idea of [46]. It consists in a refinement that allows us to relax the hypotheses extremely. This apparently new technique should be able to apply readily to the cooperative case.

Theorem 0.3 sharpens all previous hypotheses concerning to the nonlinear term of the equation of (0.6), ever in the autonomous case, which is the result given by Corollary 0.4. The assumptions we have made on∂Ωare rather general, but the ones made in [58, 59] are much weaker. On the other hand, we think that the technique developed in the proof of Theorem 0.3 might be adapted to cover more general cases, including non smooth domains or cooperative systems, but this is something we plan to do in another work.

The main difficulty of the proof of Theorem 0.6 is due to the presence on an arbitrary number of equations in (0.1). In our context we have ascertained the blow-up rates for the potential case, which is an important case, tough restricted. A future improvement may be reached by considering heterogeneous terms with non potential behavior on the boundary, in the spirit of [45], or even general nonlinearities. Of course, this is a truly non trivial problem! Going back to our result, the most important idea behind the proof is,

18 SUMMARY

probably, that we have studied the blow-up rates of (0.1) keeping in mind the uncoupled ones. We think this idea should work in larger classes of cooperative-type systems, like the

n-equations counterpart of the one considered in [31].

Probably the best uniqueness theorem that one should expect is the following one for the single equation.

Conjecture.The problem

   ∆u=a(x)f(u) in Ω u= +∞ on ∂Ω, a>0,

with∂Ω∈ C2_{, possesses a unique positive solution if and only if (0.5) has a unique positive}

solution for allT >0

This would show thata(x)does not play any important role in the uniqueness of the large solution of the equation.

Resumen

0.4

Introducci´on, motivaci´on y metodolog´ıa

El objetivo principal de esta tesis doctoral es establecer resultados de unicidad de soluci´on, tan generales como sea posible, para el siguiente problema de contorno singular el´ıptico de tipo cooperativo,        −∆ui =λiui+ n X j=1,j6=i aijuj−ai(x)fi(ui)ui en Ω, ui= +∞ en ∂Ω, 1≤i≤n, (0.19)

donden∈N,Ωes un subdominio acotado deRN,N ∈N, cuya frontera es de claseC2+ν

para ciertoν ∈(0,1),λi∈R,aij >0representan los par´ametros de acople yai ∈ Cν( ¯Ω), cona_i(x) > 0 para todo x ∈ Ω y 1 ≤ i, j ≤ n. Las condiciones de frontera deben entenderse como

lim

dist(x,∂Ω)↓0ui(x) = +

∞, 1≤i≤n,

de ah´ı que sea habitual llamar a las soluciones de (0.19)largas, oexplosivas. El modelo representado en (0.19) es una generalizaci´on de la ecuaci´on log´ıstica-difusiva con t´ermino heterog´eneo estudiada en los cap´ıtulos 6, 7 y 8 de [48], para cubrir el caso de un sistema con acoplamiento lineal de tipo cooperativo, e.d. conaij > 0para todo1 ≤i, j ≤ n. El caso en que (0.19) tiene una ´unica ecuaci´on ha sido ampliamente estudiado en la literatura. Dicho caso se remonta a los resultados pioneros de L. Bieberbach [9] y H. Rademacher [69], que consideran la ecuaci´on∆u= eu en dos y tres dimensiones, respectivamente, y los de J. B. Keller [35] y R. Osserman [66], que estudian la ecuaci´on ∆u = f(u) para cierto tipo de operadores mon´otonos. Desde entonces, los diferentes estudios sobre soluciones de ecuaciones con explosi´on en la frontera han seguido diversos caminos. Por ejemplo, estableciendo la existencia o la unicidad para t´erminos no lineales m´as generales, como en [8, 19, 28, 14], para modelos con t´erminos espaciales, [73, 7, 18, 12, 13, 41, 45], para operadores el´ıpticos m´as generales, [73, 16] o incluso considerando dominios con frontera irregular, [58, 59]. Otra rama muy estudiada es la de establecer la expansi´on asint´otica en la frontera de la soluci´on explosiva, como en [2, 14, 3]. Tambi´en hay resultados sobre multiplicidad de soluci´on en el contexto de problemas superlineales indefinidos, [42, 60,

20 RESUMEN

55, 56]. Sin embargo, la literatura existente para sistemas cooperativos con anterioridad a nuestro trabajo es realmente escasa. En efecto, un trabajo pionero en el uso de sistema cooperativos como el de (0.19) es [61, 62, 63], donde se establece la existencia y unicidad de soluciones para un sistema de dos ecuaciones con condiciones Dirichlet homog´eneas en la frontera. Uno de los primeros trabajos sobre soluciones largas en sistemas es [32], donde se estudia el modelo simbi´otico de Lotka-Volterra. Tambi´en se puede encontrar el m´as reciente [31], que estudia la existencia y unicidad de un sistema de tipo cooperativo de reacci´on difusi´on, aunque en el casoaut´onomo. Hasta donde sabemos, el primer art´ıculo que prueba la existencia de soluci´on en (0.19), para el cason = 2, es [4]. As´ı, cuando comenzamos a estudiar el modelo (0.19), el problema de la unicidad permanec´ıa ampliamente abierto, incluso en el caso en que (0.19) se reduce a una ´unica ecuaci´on.

Desde una perspectiva matem´atica, el problema de estudiar la unicidad de soluci´on en (0.19) tiene un gran inter´es en s´ı mismo. No obstante, podemos dar una motivaci´on que parte del contexto de Din´amica de Poblaciones. Supongamos queD ⊂_RN_{,N ≤3, es un} dominio acotado donde viven los individuos de una especie. Seau = u(t, x) la densidad de poblaci´on de la especie en cuesti´on, que ser´a una funci´on positiva del espacio,x∈ D, y del tiempo,t≥0. Efectuemos ahora las siguiente hip´otesis:

• La especie est´a dividida enngrupos,u1, . . . , un, de forma que cada grupo coopera con todos los dem´as, en el sentido de que el crecimiento de un grupo beneficia al crecimiento de los dem´as grupos.

• Cada grupouise esparce aleatoriamente por el h´abitatD, con tasa de difusi´ondi.

• Existen enDzonas con infinidad de recursos y zonas con recursos limitados. Esto es, en t´erminos de Din´amica de Poblaciones, donde los recursos son ilimitados la especie experimenta un crecimiento malthusiano, mientras que el crecimiento es de tipo log´ıstico si los recursos son finitos. Por simplicidad, supondremos queΩ ⊂D¯

es la zona que tiene recursos limitados.

Teniendo en cuenta todo lo anterior, un posible modelo matem´atico es el siguiente,

             ∂ui ∂t −di∆ui=λiui+ n X j=1,j6=i aijuj−bi(x)fi(ui)ui x∈ D, t >0 ui= 0 en ∂D, u(0) =u0>0, (0.20) donde1≤i≤nyb_i∈ Cν_{( ¯D)satisface}    bi(x)>0 si x∈Ω, b_i(x) = 0 si x6= Ω, 1≤i≤n.

0.4. INTRODUCCI ´ON, MOTIVACI ´ON Y METODOLOG´IA 21

De esta manera, conseguimos combinar enDlas leyes de crecimiento mencionadas arriba. Si hemos impuesto condiciones de frontera Dirichtlet homog´eneas en (0.20) es ´unicamente por simplicidad; un problema muy interesante ser´ıa considerar otras condiciones de fron-tera.

En este punto, es natural preguntarse por el comportamiento asint´otico de la ´unica soluci´on global de (0.20). La respuesta en muchos casos viene determinada por el estu-dio del problema (0.19). Suponiendo que los par´ametrosλise encuentran dentro del rango apropiado, y llamandoLminyLmaxa las solucionesminimalymaximalde (0.19), se tiene que

Lmin≤lim inf

t→+∞u(t)≤lim sup_t→+∞ u(t)≤L

max_{, x∈Ω, (0.21)}

mientras queu(t) ↑ +∞ uniformemente en subconjuntos compactos deD \Ω. Esto es,

lasmetasoluciones gobiernan la din´amica de (0.20). ´Estas son, grosso modo, soluciones

est´aticas de (0.20) que valen infinito en una zona de medida positiva; v´ease [33, 4] o [48, Chapter 5] para una definici´on rigurosa de metasoluci´on. As´ı, si (0.19) tiene una ´unica soluci´on,Lmin = Lmax en (0.21), por lo que podemos conocer el comportamiento de las soluciones de (0.20) para tiempos grandes.

La t´ecnica fundamental para obtener unicidad en (0.19) es el siguiente principio de comparaci´on, derivado de teorema de caracterizaci´on del principio del m´aximo, [54]. Sean

w1, w2 ∈

C2+ν_(∂Ω)n

tales quew1,i< w2,ipara cada1≤i≤n. Entonces, si para cada

k= 1,2 llamamosθ_[Ω,wk]a la ´unica soluci´on de

       −∆ui=λiui+ n X j=1,j6=i aijuj−ai(x)fi(ui)ui en Ω, ui=wk,i >0 en ∂Ω, 1≤i≤n, (0.22)

se tiene queθ[Ω,w1] < θ[Ω,w2]. El mismo resultado se cumple siw2 = (+∞, . . . ,+∞);

basta con aplicar el resultado anterior al dominio

Ωε :={x∈Ω : dist(x, ∂Ω)> ε}

para ε suficientemente peque˜no y hacer ε ↓ 0. Aplicando con ingenio esta t´ecnica de comparaci´on hemos obtenido buena parte de los resultados aqu´ı reunidos. Concretamente, los resultados de los cap´ıtulos 2, 3 y 4 se deben enteramente a este principio, a la vez que los resultados del cap´ıtulo 5 tamb´ıen lo utilizan.

Tambi´en hemos aplicado, en el ´ultimo cap´ıtulo de la tesis, la t´ecnica m´as frecuente en la literatura: hallar f´ormulas expl´ıcitas para lastasas de explosi´onen la frontera de las soluciones de (0.19). De hecho, este cap´ıtulo adapta la t´ecnica de localizaci´on introducida en [41] al sistema (0.19).

22 RESUMEN

0.5

Contenido

Los resultados establecidos en esta tesis doctoral han sido hallados por el autor y su director de tesis durante los a˜nos 2015, 2016 y los tres primeros meses de 2017. De estos resultados, ya han sido publicados los reunidos en [49, 50, 51], los de [52] se encuentran en proceso de publicarse y los de [53] han sido recientemente enviados a publicar. Para presentarlos en esta tesis hemos elegido un orden distinto al cronol´ogico, clasificando los resultados en dos partes, en funci´on de si estudian (0.19) en el caso particularn= 1o en el caso general. Las aportaciones principales de cada cap´ıtulo son:

(1) El primer cap´ıtulo contiene los resultados de [53]. En ´el se introduce el estudio de soluciones largas a trav´es del modelo m´as sencillo posible, el obtenido al reducir (0.19) a una ecuaci´on aut´onoma en una dimensi´on,

   u00=f(u) t∈(−T, T), u(±T) = +∞. (0.23)

La ventaja que tiene este modelo es que podemos estudiar sus soluciones existentes sin imponer de entrada ninguna condici´on sobref, al contrario que todos los estudios disponibles, que imponen como m´ınimo que f(u) seapositiva parau grande, e.d. que exista M > 0 tal que f(u) > M ≥ 0 para todo u > M. Esta diferencia hace que podamos tener comportamientos novedosos. Por ejemplo, la funci´on que asigna a cada soluci´on el tiempo m´aximal de existencia, que cuando es finita viene representada por T(x) := √1 2 Z +∞ x du q Ru x f , x:=u(0),

no tiene por qu´e cumplir que

lim inf

x→+∞T(x) = 0,

al contrario que en el caso en que f es positiva para u grande. Tamb´ıen propor-cionamos un resultado de multiplicidad de soluciones largas para cualquierf positiva que satisface la condici´on de Keller–Osserman, rompiendo la monoton´ıa en subcon-juntos de medida arbitrariamente peque˜na.

(2) El cap´ıtulo 2 contiene los resultados reunidos en [52], que consisten en tres teoremas de unicidad de soluciones largas para la ecuaci´on log´ıstica difusiva sublineal

−∆u=λu−a(x)f(u).

El primero de ellos establece unicidad en dominios estrellados cuando λ ≥ 0 y el t´ermino no lineal,f(u), cumple la siguiente condici´on,

0.5. CONTENIDO 23

adem´as de una condici´on de decrecimiento del t´ermino heterog´eneo, a(x), en un entorno de la frontera. El segundo teorema de unicidad aprovecha la misma idea de la demostraci´on del primero para generalizar el resultado, cuandoλ= 0, a una clase de dominios m´as general. Se trata de la clase formada por los dominios que admiten la representaci´on Ω = Ω0\ k [ i=1 ¯ Ωi, Ω¯i ⊂Ω0, Ω¯i∩Ω¯j =∅, 1≤i, j≤k∈N,

donde cadaΩies un dominio estrellado. El ´ultimo resultado del cap´ıtulo es de natu-raleza ligeramente distinta a los anteriores. En ´el se prueba la unicidad en dominios cuya frontera es regular cuandoλ < λ1[−∆,Ω],a(x)tiene un cierto decaimiento en

∂Ωyf(u)essuperaditivacon constanteC ≥0, e.d.

∃C≥0 : f(a+b)≥f(a) +f(b) a, b≥0.

Esta propiedad de superaditividad se remonta al Teorema 0.3 de [58], donde Marcus y V´eron la utilizan para obtener unicidad de soluciones largas de la ecuaci´on

−∆u+f(u) = 0 (0.25)

en dominios cuya frontera es localmente el grafo de una funci´on continua. ´Esta es, en efecto, una hip´otesis en la regularidad de∂Ωmucho m´as relajada que la nuestra, aunque, por otra parte, su prueba requiere quef sea una funci´on convexa. De hecho, en (0.25) no hay t´ermino lineal, y no se puede a˜nadir un t´ermino heterog´eneo de manera sin hacer alg´un trabajo adicional.

(3) En el tercer cap´ıtulo introducimos el problema de contorno singular (0.19) y damos un esquema de la demostraci´on de la existencia de soluciones minimal y maximal.

(4) El capitulo 4 expone el resultado principal de [49], donde se estudia el problema de unicidad para un sistema cooperativo con simetr´ıa radial, e.d. cuando

a_i(x) =a_i(|x|), 1≤i≤n,

y Ωes una bola o un anillo. Se prueba la unicidad cuando para cada 1 ≤ i ≤ n,

λi ≥0,ai(|x|)es una funci´on no creciente ygi(u) :=fi(u)usatisface la propiedad (0.24). En la demostraci´on se utiliza ´unicamente el principio del m´aximo, sin aludir a las tasas de explosi´on en la frontera. Es el primer teorema que se ha publicado de unicidad de soluciones largas para un sistema denecuaciones.

(5) Finalmente, el capitulo 5 est´a dedicado al estudio de las tasas de explosi´on de las soluciones de (0.19). Estos resultados fueron hallados primero para el caso especial

n = 2, [50], y despu´es para el caso general, [51]. En concreto, hallamos las tasas de explosi´on cuando, para cada 1 ≤ i ≤ n, fi(u) = upi−1 para alg´un pi > 1 y

24 RESUMEN

ai(x)tiene un decaimiento tipo potencial en la frontera deΩ,no necesariamente con

tasa fija, en el sentido de que existenb_i, γi ∈ C(∂Ω), tales quebi(z) >0para cada

z∈∂Ωyγi ≥0en∂Ω, de forma que lim x→z x∈Ω,z∈∂Ω ai(x) bi(z)[dist(x, ∂Ω)]γi(z) = 1, 1≤i≤n. (0.26)

El caso n = 1 es bien conocido desde [41]. Sin embargo, para sistemas tipo co-operativos los ´unicos resultados que hab´ıa disponibles cuando comenzamos nuestra investigaci´on eran los de [31]. El teorema principal del cap´ıtulo 5 establece que para cadaz ∈ ∂Ωexistenαi(z), Ai(z) > 0tales que para cualquier soluci´on de (0.19),

u= (u1, . . . , un), se tiene que lim x→z x∈Ω ui(x) dist(x, ∂Ω)−αi(z) =Ai(z), 1≤i≤n.

De forma m´as precisa, lo que dice el teorema es que las tasas de explosi´on de (0.19) se puede establecer en funci´on de c´omo est´en de cercanas entre si las tasas de explosi´on

delsistema desacoplado, que es el obtenido cuandoaij = 0para todo1 ≤i, j ≤n.

Lo m´as sorprendente es que si todas las tasas de explosi´on del sistema desacoplado est´an suficientemente cercanas entre s´ı, las tasas de explosi´on del correspondiente sistema acoplado son las mismas, independientemente del tama˜no de los t´erminos de acople,aij. El teorema tambi´en determina c´omo cambia la naturaleza de las tasas de explosi´on de (0.19) cuando alguna de las tasas de explosi´on del sistema desacoplado se aleja de las dem´as. Como nuestro modelo es heterog´eneo, las tasas de explosi´on del sistema desacoplado pueden variar en funci´on del puntoz∈∂Ω, debido a (0.26). Por tanto, las tasas de explosi´on se pueden comportar en unas zonas de∂Ωcomo si el

sistema estuviera desacopladoy en otras zonas de∂Ωverse afectadas por los t´eminos

de acopleaij. Este comportamiento qued´o registrado por primera vez en [50]. M´as a´un, ning´un art´ıculo previo a [51] estudia las tasas de explosi´on de un sistema de n

ecuaciones.

0.6

Conclusiones

Pese a que los problemas con condiciones singulares en la frontera tienen una gran difi-cultad, hemos podido obtener una amplia gama de teoremas de unicidad, fundamentalmente usando el teorema de comparaci´on descrito en 0.4. Que el sistema sea de tipo cooperativo es algo crucial en el an´alisis realizado. En efecto, si alguno de losaij de (0.19) fuese negativo no tendr´ıamos disponible el teorema de caracterizaci´on del principio del m´aximo [54], por lo que fallar´ıan las t´ecnicas de comparaci´on, que son la herramienta fundamental de esta tesis. Claro que, por otra parte, ser´ıa muy interesante considerar este tipo de situaciones en el futuro. Seguramente un buen acercamiento ser´ıa a trav´es del caso en que (0.19) es cuasi-cooperativo, e.d. cuandon= 2ya12a21 >0, ya que en este caso tambi´en est´a disponible

0.6. CONCLUSIONES 25

Como resultado de nuestra investigaci´on, hemos refinando y extendido las t´ecnicas de unicidad m´as habituales, adem´as de aportar algunas ideas nuevas. El resultado descrito en

(4)es una adaptaci´on del resultado de [46] para el sistema cooperativo. Esta adaptaci´on, m´as inmediata para el caso de la bola, requiere una construcci´on auxiliar mucho m´as com-pleja para el caso del anillo. Asimismo, los dos primeros resultados resumidos en(2) se deben a un refinamiento extremo de la idea fundamental de [46], que nos permite relajar enormemente las hip´otesis. Esta nueva t´ecnica se deber´ıa poder usar para extender estos resultados al caso de un sistema cooperativo sin presentar demasiadas complicaciones.

El ´ultimo resultado resumido en(2)es de una generalidad asombrosa, pues, como se ver´a en el cap´ıtulo 2, la propiedad de superaditividad def relaja todas las hip´otesis que se han hecho previamente, incluso en el caso aut´onomo. En cuando a las hip´otesis que hemos pedido a ∂Ω, aunque sean bastante generales, no son tan d´ebiles como las realizadas en [58, 59]. Por otro lado, creemos que las t´ecnicas desarrolladas en nuestra prueba podr´ıa adaptarse para cubrir casos m´as generales, como dominios cuya frontera no sea necesaria-mente diferenciable, o sistemas cooperativos. Pero eso es algo que haremos en otro trabajo. Respecto al ´ultimo de nuestros teoremas, descrito en(5), quis´ıeramos destacar la difi-cultad de determinar las tasas de explosi´on para una cantidad arbitraria de ecuaciones. En nuestro caso, hemos logrados dar con las f´ormulas para el caso concreto en el que tanto los t´erminos no lineales como los factores espaciales se comportan como potencias. Una posible generalizaci´on, nada obvia, ser´ıa hallar las tasas prescindiendo de alguna de estas hip´otesis. Seguramente la idea m´as importante para obtener nuestro resultado haya sido pensar en las tasas de explosi´on de (0.19) en funci´on de las tasas del sistema desacoplado. Esta forma de proceder podr´ıa intentar aplicarse a otro tipo de sistemas cooperativos, con diferente acople, como el equivalente denecuaciones al estudiado en [31].

Para terminar este resumen, dejamos el enunciado de lo que creemos que deber´ıa ser el teorema ´optimo de unicidad para la ecuaci´on.

Conjetura.El problema    ∆u=a(x)f(u) en Ω u= +∞ en ∂Ω,

con∂Ω ∈ C1 _{tiene una ´unica soluci´on si y s´olo si (0.23) admite una ´unica soluci´on para}

todoT >0.

Part I

Large solutions for the equation

Chapter 1

Multiplicity of large solutions in one

spatial dimension

The main goal of this chapter is to analyze the existence, uniqueness and multiplicity of positive solutions of the singular boundary value problem

   u00=f(u), t∈[0, T], u0(0) = 0, u(T) = +∞, (1.1)

whereT ∈ (0,∞)andf ∈ C1_{[0,+∞)satisfiesf(0) = 0. So,0is a constant solution of} u00 = f(u). By reflection aroundt= 0, these solutions provide us with the positive large solutions of the singular problem

 



u00=f(u), t∈[−T, T], u(−T) =u(T) = +∞.

By a positive large solution of (1.1) it is meant any positive solution in[0, T)such that

lim

t↑T u(t) = +∞.

The singular problem (1.2) has been widely studied in the literature. However, almost all available results assumed f ≥ 0 in[0,∞), or, at least, for sufficiently large u, [14, 15]. Here we are not imposing any special restriction on the sign off.

As for arbitraryf(u)the existence and multiplicity of positive solutions of (1.1) might depend on the length of the interval,T > 0, it is very natural to analyze the existence of positive explosive solutions of the associated Cauchy problem

   u00 =f(u), u(0) =x >0, u0(0) = 0, (1.2) 29

30CHAPTER 1. MULTIPLICITY OF LARGE SOLUTIONS IN ONE SPATIAL DIMENSION

wherex >0is regarded as parameter.

Although there is a huge interest in analyzing the existence and the uniqueness of large solutions for wide classes of singular sublinear boundary value problems, because they pro-vide us with the limiting profiles as time grows of the solutions of large classes of diffusive logistic equations of degenerate type, where the species can growth exponentially in some protection zones of the territory, [33, 34, 18, 12, 13, 48], and the classical condition of J. B. Keller [35] and R. Osserman [66], as well as some of their variants, as those of [40] and [19], have dominated the scenario of this theory during the last two decades, except in Section 1.1 of [48], no serious effort has been made to realize the true meaning of the several Keller–Osserman conditions involved.

In most of the literature collected in our bibliography, the Keller–Osserman condition is imposed in order to guarantee the existence of large solutions of some autonomous or non-autonomous problem where the nonlinearity uses to be chosen so that the underlying semilinear elliptic equation can exhibit, at most, a unique large solution; the main aim of most of these papers being to show that any large solution must have the same blow-up rate on the edges of the domain to infer from this feature the uniqueness of the large solution by means of a rather standard comparison device. As a consequence of this severe focusing of most of experts’s attention, the real meaning of the so called Keller–Osserman condition remains a true enigma!

This prompted us to focus attention in the simplest autonomous one dimensional sin-gular problem (1.1) in order to characterize, simply, the values ofT for which this singular problem admits a positive solution. Should it be the case, our second aim being either estab-lishing uniqueness, or multiplicity results, keeping in mind, rather crucially, that, in general, (1.1) might admits positive solutions for some range of values ofTbut not for others! This apparently new methodology contrasts heavily with most of the available results in the lit-erature, where the Keller–Osserman condition entails the existence of a positive solution of the singular problem (1.1) for everyT > 0, because the functionf(u) is required to sat-isfy some additional monotonicity property to infer from it the uniqueness of the positive solution of (1.1). So, our methodology here seems completely new.

In the context of superlinear indefinite problems there are available some multiplicity results, as [29, 42, 60], but probably the most astonishing existing multiplicity results are those of [56] and [55], where it was established that ifa(x)changes of sign in the interval [−T, T], then the problem

 



−u00=λu−a(x)up, in [−T, T], u(−T) =u(T) = +∞,

wherep > 1, can admit an arbitrarily large number of positive solutions by takingλ < 0 sufficiently large. In these results the multiplicity is caused by the fact thata(x)changes of sign andλ < 0is very large, and is far from attributable to the nature of the nonlinearity,

f(u) = up, with p > 1, for which the singular problem (1.1) has a unique positive solu-tion for eachT > 0. Instead, the multiplicity results of this chapter are attributable to the

1.1. THE ASSOCIATED CAUCHY PROBLEM 31

oscillatingproperties of the functionf(u)in (1.1), even whenf(u)>0for allu >0. Con-sequently, our findings here are of a great novelty and independent of all previous available multiplicity results.

Besides the introduction, this chapter consists of 3 sections. Section 1.1 studies the Cauchy problem (1.2), Section 1.2 deals with the existence, uniqueness and multiplicity of positive solutions for the singular problem (1.1), and Section 1.3 gives some interest-ing counterexamples to an important result of [19]. Astonishinterest-ingly, our main multiplicity result in Section 1.2 shows how destroying the monotonic character of any increasing func-tionf(u)on a compact set with arbitrarily small measure can originate an arbitrarily large number of positive solutions for the singular problem (5.1).

1.1

The associated Cauchy problem

Sincef ∈ C1_{[0,+∞)andx > 0, by the main existence theorem forC}1 _{nonlinearities, it}

becomes apparent that, for everyx >0, the initial value problem (1.2) possesses a maximal positive solution, u(t), t ∈ [0, Tmax(x)), for some Tmax(x) ∈ (0,+∞]. Moreover, the

following result holds.

Theorem 1.1 The following properties are satisfied:

(a) Iff(x) = 0, thenxis a constant solution and hence,Tmax(x) = +∞.

(b) Iff(x)>0andTmax(x) = +∞, then, eitheru(t)is periodic, or

lim

t↑+∞u(t) =ω for someω > xsuch thatf(ω) = 0, or

lim

t↑+∞u(t) = +∞.

(c) Iff(x)>0andTmax(x)<+∞, thenu0(t)>0for allt∈[0, Tmax(x)),

lim t↑Tmax(x)

u(t) = +∞, (1.3)

Rθ

x f >0for allθ > x, and

Tmax(x) = 1 √ 2 Z +∞ x dθ q Rθ x f <+∞. (1.4)

(d) Iff(x) <0andTmax(x) = +∞, then, eitheru(t)is periodic, oru0(t) <0for all t >0and there existsα∈[0, x)such thatf(α) = 0and

lim

32CHAPTER 1. MULTIPLICITY OF LARGE SOLUTIONS IN ONE SPATIAL DIMENSION

(e) If f(x) < 0 and Tmax(x) < +∞, then u0(t) < 0 for all t ∈ (0, Tmax(x)] and u(Tmax(x)) = 0.

Proof: Iff(x) = 0, thenxis a constant solution of (1.2) and hence,Tmax(x) = +∞. In

particular,u(t)is periodic. This proves Part (a). Now, suppose thatf(x)>0. Then,

u00(0) =f(u(0)) =f(x)>0

and hence, sinceu0(0) = 0, there exists δ > 0 such that u0(t) > 0 for all t ∈ (0, δ). Eitheru0(t)>0for allt ∈(0, Tmax(x)), or there existst0 >0such thatu0(t) >0for all t∈(0, t0)andu0(t0) = 0. In the second case, by reflectingu(t)aboutt=t0, it becomes

apparent thatTmax(x) = +∞and thatu(t)is a nontrivial periodic solution ofu00 =f(u).

Suppose

u0(t)>0 for all t∈(0, Tmax(x)) (1.5)

and, in addition,Tmax(x) = +∞. Then, by (1.5),

lim

t↑+∞u(t) =ω ∈(0,+∞]

is well defined. Moreover,f(ω) = 0ifω <+∞, because

0 = lim t↑+∞u

00

(t) = lim

t↑+∞f(u(t)) =f(ω),

which concludes the proof of Part (b).

Suppose (1.5) andTmax(x)<+∞. Then,

lim sup T↑Tmax(x)

u(t) +u0(t)=∞. (1.6)

Moreover, for eacht∈(0, Tmax(x)), integrating the differential equation yields

u0(t) =

Z t

0

f(u(s))ds.

Thus, if there is a constantCsuch thatu(s)≤Cfor alls∈[0, Tmax(x)), then

|u0(t)| ≤Tmax(x) max

u∈[0,C]|f(u)|<+∞,

which contradicts (1.6). Therefore,

lim t↑Tmax(x)

u(t) = +∞,

which provides us with (1.3). Finally, multiplyingu00=f(u)byu0and integrating in[0, t],

t < Tmax(x), we obtain that

1 2(u

0_(t))2₌

Z u(t)

x

1.1. THE ASSOCIATED CAUCHY PROBLEM 33

Sinceu(t)ranges in[x,+∞)ast∈ [0, Tmax(x)), (1.5) and (1.7) imply that

Rθ x f > 0for allθ > x. Moreover, Tmax(x) = Z Tmax(x) 0 dt= Z Tmax(x) 0 u0(t) q 2Ru(t) x f(s)ds dt= Z +∞ x dθ q 2Rθ x f ,

which establishes (1.4) and ends the proof of Part (c).

Finally, supposef(x)<0. Then, sinceu00(0) =f(x)<0, there existsδ >0such that

u0(t) < 0for allt ∈ (0, δ). If there existst0 > 0 such thatu0(t) < 0for allt ∈ (0, t0)

andu0(t0) = 0, thenu(t)is periodic and henceTmax(x) = +∞. Therefore, ifu(t)is not

periodic andTmax(x) = +∞, thenu0(t) < 0for allt > 0and hence, Part (d) holds. If Tmax(x) < +∞, necessarilyu(Tmax(x)) = 0. Moreover,u0(Tmax(x))< 0, because if it

vanishes, thenu≡0, which is impossible. This ends the proof. ₂

By Theorem 1.1, if there existsT > 0 for which (1.1) possesses a positive solution, then, settingx:=u(0), we have thatx >0,f(x)>0,R_xθf >0for allθ > x, and

T = √1 2 Z +∞ x dθ q Rθ x f =Tmax(x).

Moreover, the following converse holds.

Lemma 1.2 Letx > 0 be such thatf(x) > 0andR_xθf > 0for allθ > x. Then, (1.5)

holds. If, in addition,

1 √ 2 Z +∞ x dθ q Rθ x f <+∞,

then the unique solution of the Cauchy problem(1.2)blows up at

Tmax(x) = 1 √ 2 Z +∞ x dθ q Rθ x f .

Therefore, the singular problem(1.1)admits a positive solution ifT =Tmax(x).

Proof: Since f(x) > 0, by continuity, there existsδ > 0such thatu00(t) = f(u(t)) >0 for allt∈(0, δ). Hence,u0is increasing(0, δ). So, sinceu0(0) = 0, we find thatu0(t)>0 for allt∈(0, δ). Consider

ˆ

δ := sup{δ >0 : u0(t)>0 for all t∈(0, δ)}.

Necessaryδˆ=Tmax(x), because, otherwise, we may infer from

1 2(u 0_(t))2 ₌ Z u(t) x f(s)ds for every t∈(0,δˆ),

34CHAPTER 1. MULTIPLICITY OF LARGE SOLUTIONS IN ONE SPATIAL DIMENSION that 0 = 1 2(u 0 (ˆδ))2 = Z u(ˆδ) x f(s)ds,

which contradicts the assumption thatR_xθf >0for allθ > x, sinceu(ˆδ)> x. Thus,

u0(t)>0 for all t∈(0, Tmax(x)),

which is the first assertion of the lemma. Thanks to (1.5), for everyt∈ (0, Tmax(x)), we

find that t= Z t 0 ds= Z t 0 u0(s) q 2R_xu(s)f ds = √1 2 Z u(t) x dθ q Rθ x f < √1 2 Z +∞ x dθ q Rθ x f <+∞.

Therefore, lettingt↑Tmax(x)yields

Tmax(x)≤ 1 √ 2 Z +∞ x dθ q Rθ x f <+∞.

Consequently, by Part (c) of Theorem 1.1, the solution blows up atTmax(x)and (1.4) holds.

This ends the proof. ₂

According to these results, in searching for the solutions of the singular problem (1.1), it is natural to consider the set

D:= x >0 : f(x)>0 and Z θ x f >0 for all θ > x ,

as well as the operatorT :D →(0,+∞]defined by

T(x) := √1 2 Z +∞ x dθ q Rθ x f(s)ds (1.8)

for allx∈ D. Indeed, in terms of(D,T), the next result holds.

Theorem 1.3 The singular boundary value problem(1.1)possesses a positive solution if,

and only if, there existsx∈ Dsuch thatT =T(x). Moreover, in such case

T =T(x) =Tmax(x)<+∞,

where Tmax(x) stands for the blow-up time of the solution of the Cauchy problem (1.2).

Furthermore, the number of positive solutions of (1.1),n(T), is given by

1.1. THE ASSOCIATED CAUCHY PROBLEM 35

Similarly, if T(x) = +∞for somex∈ D, thenTmax(x) = +∞. In particular, the solution

of the Cauchy problem(1.2)cannot provide us with a solution of the singular problem(1.1)

for someT >0.

Proof: GivenT >0, suppose thatT = T(x)for somex ∈ D. Then, by Lemma 1.2, the unique solution of the Cauchy problem (1.2),u(t), satisfies (1.5) and

lim

t↑T u(t) = +∞, T =Tmax(x) =T(x).

Therefore,u(t)provides us with a positive solution of the singular problem (1.1).

Conversely, suppose that (1.1) admits a positive solution, u(t), and setx := u(0). If

x = 0, thenu ≡ 0, which contradicts our assumption. Thus,x > 0and it follows from Theorem 1.1 thatT =Tmax(x) =T(x).

The number of positive solutions of (1.1) equalsn(T)because one can establish a bi-jection between the solutions of the singular problem (1.1) and its values att = 0, by the uniqueness of the solution of the initial value problem (1.2).

Finally, suppose thatT(x) = +∞for somex∈ Dand letu(t)be the unique solution of the Cauchy problem (1.2). Sincex∈ D, by the first statement of Lemma 5.26,u0(t)>0 for allt∈(0, Tmax(x)). Hence,

t= √1 2 Z t 0 u0(s) q Ru(s) x f = √1 2 Z u(t) x dθ q Rθ x f (1.9)

for allt∈(0, Tmax(x)). SupposeTmax(x)<+∞. Then, by (1.3),

lim t↑Tmax(x)

u(t) = +∞

and hence, lettingt↑Tmax(x)in (1.9) yields

Tmax(x) =T(x) = +∞

which contradictsTmax(x) <+∞. Therefore,Tmax(x) = T(x) = +∞, which ends the

proof. ₂

Remark 1.4 Suppose thatf(u)≥0for allu >0. Then,_dθd Rθ

x f =f(θ)≥0for allθ >0 and hence,R_xθf > 0 for allθ > xprovided f(x) > 0. Thus, D = (0,+∞)\f−1(0), though T(x) might be finite or infinity. Therefore, in this important case, the singular problem (1.1) admits a positive solution for someT >0if, and only if, there existsx >0 such thatf(x)>0andT(x) =T. Moreover,n(T)equals the number of suchx’s.

Naturally, we can extend the definition ofT by setting

T(x) =Tmax(x) for all x∈(0,+∞)\ D.

According to Theorem 1.3, this implies thatT ≡ Tmax in(0,∞). The following result

36CHAPTER 1. MULTIPLICITY OF LARGE SOLUTIONS IN ONE SPATIAL DIMENSION

Lemma 1.5 Tmax(x) = +∞ifx∈(0,+∞)\ Dwithf(x)≥0.

Proof: Letx > 0 be such that x /∈ D. If f(x) = 0, thenx is a constant solution of

u00 = f(u) and henceTmax(x) = +∞. So, supposef(x) > 0. According to Part (c) of

Theorem 5.25,Tmax(x)cannot be finite, because in that case we should have

Rθ

x f >0for allθ > x, and hence,x∈ D, which contradices our assumption. Therefore, in all possible cases,Tmax= +∞. The proof is complete. 2

By the theorem of differentiability of Peano,Tmax(x), and hence the extended function

T(x), is continuous with respect tox ∈(0,+∞). Therefore, as soon asT(x0)<+∞for

somex0∈ D, there exists an open subinterval(a, b)⊂ D, maximal for the inclusion, such

thatx0∈(a, b),T(a) =T(b) = +∞, and

T(x)<+∞ for all x∈(a, b).

1.2

Existence, uniqueness and multiplicity

As the graph of the time mapT can be as wiggle as we wish by choosing an appropriate

f(u), it is a challenge to analyze the general global behavior ofT, unless we impose some additional (severe) restrictions on f(u), like the monotonicity of f(u). The next result explains why.

Theorem 1.6 Suppose that there existsx0 ≥0, withf(x0) >0andT(x0) < +∞, such thatf(u)is increasing foru > x0. Then,T(x)is decreasing for x > x0. In particular,

T(x)<T(x0)<+∞for allx > x0. Moreover,

lim

x↑+∞T(x) = 0.

Therefore, for everyT ∈ (0,T(x0)), the singular problem(1.1) possesses, at least, one

positive solution.

Proof: Sincef is increasing in[x0,+∞), we have thatf(x) > f(x0) >0for allx > x0

and thatRθ

x f >0for allθ > x. Thus,[x0,+∞)⊂ D and hence, for everyx ≥x0,T(x) is given through (1.8). Consequently, performing the change of variableτ := θ−x, we find that, for everyx≥x0,

T(x) = √1 2 Z +∞ 0 dτ q Rx+τ x f(s)ds = √1 2 Z +∞ 0 dτ q Rτ 0 f(x+t)dt .

Supposex0 ≤x < y. Then,f(x0+t)≤f(x+t)< f(y+t)for allt >0and hence,

s Z τ 0 f(x0+t)dt≤ s Z τ 0 f(x+t)dt < s Z τ 0 f(y+t)dt for every τ >0.

1.2. EXISTENCE, UNIQUENESS AND MULTIPLICITY 37

SinceT(x0)<+∞, the involved integrals are convergent. Therefore, we find thatT(x)>

T(y). The monotonicity ofT forx≥x0, entails the existence of the limit

L:= lim

x↑+∞T(x)≥0.

The following result of S. Dumont et al. [19] guarantees that actuallyL= 0.

Lemma 1.7 Suppose that there exists x0 > 0 such that f(u) ≥ 0 for all u ≥ x0, and

T(x)<+∞for somex > x0. Then,

lim inf x↑+∞ Z +∞ 0 dτ q Rτ 0 f(x+t)dt = 0. (1.10)

Finally, the last assertion of the theorem is a direct consequence from Theorem 1.3, as for everyT ∈(0, T(x0))there existsx > x0such thatT(x) =T. 2

Naturally, the next result follows easily from Theorem 1.6.

Corollary 1.8 Supposef(0) = 0,f is increasing, andT(x0) < +∞ for somex0 > 0.

Then, the singular problem(1.1)possesses a unique positive solution for eachT >0.

Proof:Sincef(0) = 0,T(0) = +∞. Moreover, by Theorem 1.6,

lim

x↑+∞T(x) = 0.

Therefore, sinceT is continuous and decreasing when it is finite, for everyT > 0 there exists a uniquex >0such thatT(x) =T. Theorem 1.3 ends the proof. ₂

Remark 1.9 In order to get the existence of a positive solution of the singular problem (1.1) for sufficiently smallT >0one should impose

lim inf

x→+∞T(x) = 0. (1.11)

Nevertheless, even when f ≥ 0 or f(u) > 0 for all u > 0, the singular problem (1.1) might admit an arbitrarily large number of positive solutions for sufficiently largeT > 0, as established by the next result.

Theorem 1.10 Letx1, . . . , xp ∈(0,+∞)be distinct andf ∈ C1[0,+∞)such that

(a) f(0) =f(xj) = 0for eachj ∈ {1, ..., p}andf(u)>0ifu /∈ {x1, ..., xp}.

(b) Setting x0 := 0 and xp+1 := +∞, for every j ∈ {0, ..., p}, there exists xj0 ∈

38CHAPTER 1. MULTIPLICITY OF LARGE SOLUTIONS IN ONE SPATIAL DIMENSION

Then, there existsT∗ > 0such that(1.1) possesses, at least2p+ 1positive solutions for

everyT > T∗. Moreover, there existsT∗ >0such that(1.1)possesses, at least, a positive solution for eachT < T∗.

If, in addition,f is increasing for sufficiently largeu, thenT∗ >0can be shortened, if necessary, so that(1.1)admits a unique positive solution for everyT < T∗.

Proof:Sincexj,0≤j ≤p, are constant solutions ofu00=f(u),

T(xj) =Tmax(xj) = +∞. (1.12)

Moreover, since on each of the intervals(xj−1, xj),1 ≤ j ≤ p+ 1, T is assumed to be somewhere finite, the following values are well defined

min x∈(xj−1,xj)

T(x)<+∞, 1≤j≤p,

and, thanks to Lemma 1.7,

lim inf

x↑+∞ T(x) = 0.

Therefore, combining the continuity ofT with (1.12), it becomes apparent that for every

T > T∗ := max

1≤j≤p x∈(minxj−1,xj)

T(x),

there exist at least2p+ 1different points,x∈ D, such thatT(x) =T. The final assertion of the theorem is a byproduct of Theorem 1.6. This ends the proof. ₂

For everyx1, . . . , xp ∈(0,+∞)withxi 6=xjifi6=j, the function

f(u) :=u

p

Y

j=1

(u−xj)2 (1.13)

satisfies all the requirements of the theorem, even the monotonicity foru > xp, and, since

lim u↑+∞

f(u)

u2p+1 = 1,

it is easily seen that

T(x)<+∞ for all x∈ D= (0,∞)\ {x1, ..., xp}.

Therefore, according to Theorem 1.10, there exist0< T∗ < T∗ <+∞such that, for this

special choice off(u), the singular problem (1.1) possesses at least2p+1positive solutions ifT > T∗and a unique positive solution ifT < T∗. Figure 1.1 shows the graph of the map

T(x)associated tof(u), given by (1.13), for the special choicep= 2,x1 = 4andx2 = 8.

RegardingT >0as a parameter we can easily ascertain the bifurcation diagram of the large positive solutions of the associated singular problem. Asf(u)is increasing for allu >8, by

1.2. EXISTENCE, UNIQUENESS AND MULTIPLICITY 39

Figure 1.1: The time mapT(x)forf(u) =u(u−4)2_(u−8)2

Theorem 1.10 there existsT∗ >such that (1.1) admits a unique positive solution for every

T < T∗. Moreover, from the proof of Theorem 1.6 we can infer thatu(0) = x ↑ +∞as

T ↓0. Hence, there exists a branch of large positive solutions that bifurcates from infinity. As we letT grow, two new branches of large positive solutions appear, just like shown by Figure 1.2, where we are plotting the parameterT in abscisas versus the initial data,x, in ordinates. This simple example provides us with a rather general scheme to generate as

Figure 1.2: Bifurcation diagram forf(u) =u(u−4)2_(u−8)2

many large positive solutions as we wish starting at an arbitrary increasing functionf(u). Indeed, fixedp≥1andpdistinct points,x1, ..., xp>0, pickη >0such that

0< xj−1−η < xj−1+η < xj−η, 2≤j ≤p, and then, changef inside each of the intervals